It turns out it does build with a newer Flex SDK, but all I get when I run the resulting .swf is a grey screen.
For what it's worth, I built it with 4.6.0 of the Flex SDK, which Adobe is still hosting: https://fpdownload.adobe.com/pub/flex/sdk/builds/flex4.6/fle...
So yeah, problem solved, I guess?
I'll see if I can give this a try this evening